SOLUTIONS FOR HOMEWORK 6
13.1. (b) We have to show that, for any > 0, there exists > 0 with
the property that n |f (bk )g(bk ) f (ak )g(ak )| < whenever ([ak , bk ])n are
k=1
k=1
non-overlapping inter
HOMEWORK 2
Problems assigned up to and including Friday, 04/08. The solutions
are due on Friday, 04/15.
Assigned Monday, 04/04:
Chapter 17: 17.4, 17.5, 17.6, 17.7, 17.12, 17.13.
Note. In Exercise 17.1
HOMEWORK 4
Problems assigned up to and including Friday, 04/29. The solutions
are due on Friday, 05/13.
Assigned Wednesday, 04/27:
Chapter 23: 23.2, 23.4, 23.12, 23.13, 23.14.
Back to the syllabus.
Ba
SOLUTIONS FOR HOMEWORK 3
17.8. Note that (a) follows from (b). Indeed, dene the measure via (E) =
(E)/(X). Then Mp (f ) = f Lp () . So, we prove (b). Throughout the rest of the
proof, (X) (0, 1]. The
HOMEWORK 5
Problems assigned up to and including Friday, 05/13. The solutions
are due on Friday, 05/20.
Assigned Friday, 05/06:
Suppose X is a Banach space, equipped with the operation of multiplicati
HOMEWORK 3
Problems assigned up to and including Friday, 04/15. The solutions are due on Friday,
04/29.
Assigned Wednesday, 04/13:
Chapter 17: 17.8.
A. Suppose X is a Banach space, and X is separable.
SOLUTIONS FOR HOMEWORK 4
23.2. Solution 1. For N N, let GN = cfw_(x, y) : y = f (x), N < x N. It suces
to show that, for any N, GN belongs to (B B), and has measure 0. Henceforth, we x
N. Our goal is
SOLUTIONS FOR HOMEWORK 5
Suppose X is a Banach space, equipped with the operation of multiplication. That is,
for x, y X, we have xy X. Throughout, we assume that multiplication is linear
that is, (1
TAKE-HOME FINAL FOR MATH 210C (SPRING 2011)
The solutions are due by 2 pm on Friday, June 3. You can either give your
papers to me directly, slide them under my door, or place them in my mailbox.
I pr
SOME PRACTICE PROBLEMS FOR THE QUALIFIER
Chapter 9: 9.14, 9.15, 9.18 9.39.
Chapter 10: 10.8.
Chapter 11: 11.4 (under the condition that (cfw_x X : f (x) 1/n) < for any n N).
Chapter 12: 12.11, 12.13.
SOLUTIONS FOR THE FINAL
1 (10 points): Prove Clarksons Inequality for 2 p < : if f, g Lp (), then
f +g
2
p
f g
2
+
p
p
f
p
p
p
+ g
2
p
p
.
You can assume that the functions are real-valued (the inequa
SOLUTIONS FOR MIDTERM
Throughout, all measure spaces are assumed to be -nite. All functions are extended real
valued.
1 (10 points): Prove the uniqueness of the weak limit. More precisely, suppose 1 p
TAKE-HOME MIDTERM FOR MATH 210C (SPRING 2011)
The solutions are due by 2 pm (that is, by the beginning of the lecture) on
Friday, May 6. You can either give your papers to me directly, slide them
unde
SOLUTIONS FOR HOMEWORK 1
16.3. It suces to consider the case of f 0. Let F = f r , G = f r , H = f (1)r ,
a = p/(r), and b = q/(r(1 ). Then 1/a + 1/b = 1, and F = GH. By Hlder
o
1/a
1/b
r
Inequality,
HOMEWORK 1
Problems assigned up to and including Friday, 04/01. The solutions
are due on Friday, 04/08.
Assigned Wednesday, 03/30:
Chapter 16: 16.3, 16.4, 16.6, 16.16, 16.17, 16.22.
Note. In Exercise
SOLUTIONS FOR HOMEWORK 5
b
b
12.2. We show that Va (1/f ) Va (f )/c2 . To this end, it suces to show that,
b
b
for any partition P = cfw_a = x0 < x1 < . . . < xn = b, Va (1/f, P) Va (f )/c2.
But
n
b
V
HOMEWORK 5
Problems assigned up to and including Friday, 02/11. The solutions
are due on Friday, 02/18.
Assigned Monday, 02/07:
Chapter 12: 12.2, 12.6, 12.10, 12.12, 12.14.
Assigned Wednesday, 02/11:
HOMEWORK 6
Problems assigned up to and including Friday, 02/25. The solutions
are due on Friday, 03/04.
Assigned Wednesday, 02/16:
Chapter 13: 13.1(b), 13.3, 13.4, 13.7, 13.8, 13.10, 13.12, 13.13(a,c)
HOMEWORK 7
Problems assigned up to and including Friday, 03/04. The solutions
are due on Friday, 03/11.
Assigned Monday, 02/28:
A. Prove that any two norms on Rn are equivalent.
Hint. Prove that any n
PRACTICE PROBLEMS FOR THE MIDTERM: THE
SOLUTIONS
The in-class midterm exam will be given on Wednesday, Feb. 2. The
material from Homeworks 1-3 (Chapters 8, 9, and 10) will be covered.
8.10. Let D0 = D
SOLUTIONS FOR HOMEWORK 7
A. Prove that any two norms on Rn are equivalent.
Hint. Prove that any norm on Rn is equivalent to
x = (x1 , . . . , xn ), x 1 = n |xi |).
i=1
1,
introduced in Section 15.II
SOLUTIONS FOR MIDTERM
1 (10 points): Suppose M > 0, (X, A, ) is a measure space, D A, and f is
an extended real valued function on D, such that
|f |p d
D
p (0, ). Show that |f | M almost everywhere on
SOLUTIONS FOR FINAL
1 (10 points): Suppose E is a subset of R of positive measure. Prove that E + E =
cfw_x + y : x, y E contains a non-trivial interval.
Hint. By Problem 13.19, there exists an interv
TAKE-HOME FINAL FOR MATH 210B (MARCH 2011)
The solutions are due by 1:30 pm on Friday, March 18. You can either give your
papers to me directly, slide them under my door, or place them in my mailbox.
SOLUTIONS FOR HOMEWORK 1
8.1. It suces to show that, for any non-negative simple function
on D, satisfying D d = +, we have sup D d = + (the
supremum runs over all -integrable simple functions , sati
HOMEWORK 1
Problems assigned up to and including Friday, 01/07. The solutions
are due on Friday, 01/14.
Assigned Wednesday, 01/05:
Chapter 8: 8.1, 8.3, 8.4, 8.5, 8.6, 8.8, 8.9, 8.11.
Back to the sylla
HOMEWORK 2
Problems assigned up to and including Friday, 01/14. The solutions
are due on Friday, 01/21.
Assigned Wednesday, 01/12:
Chapter 9: 9.7, 9.9, 9.10, 9.11, 9.12, 9.17, 9.19, 9.22, 9.34.
You ca
SOLUTIONS FOR HOMEWORK 2
9.7. (a) Let fn = fAn = f 1An . Then |fn | |f |. Moreover,
|fn | d. By Monotone Convergence Theorem,
X
|f | d =
|f | d < .
|f | d = lim
n
X
An
An
By Observation 9.2, f is -int
SOLUTIONS FOR HOMEWORK 3
A. Prove that, for any L -integrable function f on R, and any > 0, there
N
exists a step function g =
i=1 ai 1Ii (Ii are nite intervals, ai R) s.t.
|f g| dL < .
R
Hint. First
HOMEWORK 4
Problems assigned up to and including Friday, 01/28. The solutions
are due on Friday, 02/11.
Assigned Wednesday, 01/26:
Chapter 11: 11.1, 11.2(a,b), 11.3, 11.6, 11.7.
Back to the syllabus.